Poisson Distribution and Poisson Processes

Counts, event rates, and random arrivals

Statistics

Probability

Distributions

Stochastic Processes

The Poisson distribution and homogeneous Poisson process, including derivations, waiting-time relationships, assumptions, and biological interpretation.

Author

Jiahong Sun

Published

September 11, 2026

Modified

September 11, 2026

1 Topic at a Glance

One-sentence summary: The Poisson distribution models the number of events in a fixed exposure, while a Poisson process models how such events accumulate through continuous time or space at a constant rate.

Item	Description
Domain	Probability & Randomness; count distributions and point processes
Main question	How many events occur in a region, and where or when do they occur?
Support	Count $X\in\{0,1,2,\ldots\}$; process $N(t)\in\{0,1,2,\ldots\}$
Parameters	Mean count $\mu>0$; process rate $\lambda>0$ per unit exposure
Typical applications	Mutation counts, sequence starts, recombination events, failures, arrivals
Prerequisites	Bernoulli and binomial distributions; independence and conditional independence

2 The Poisson Distribution

A random variable $X$ has a Poisson distribution with mean parameter $\mu>0$ when

\[ \boxed{ P(X=x)=e^{-\mu}\frac{\mu^x}{x!}, \qquad x=0,1,2,\ldots } \]

written $X\sim\operatorname{Poisson}(\mu)$. The parameter is dimensionless: if events occur at rate $\lambda$ per unit exposure and the exposure is $t$, then $\mu=\lambda t$.

Normalization follows from the exponential series:

\[ \sum_{x=0}^{\infty}e^{-\mu}\frac{\mu^x}{x!} =e^{-\mu}e^\mu=1. \]

The probability of zero events is $e^{-\mu}$, and successive probabilities obey

\[ \frac{P(X=x+1)}{P(X=x)}=\frac{\mu}{x+1}. \]

This recursion shows that probability masses increase while $x+1<\mu$ and decrease after $x+1>\mu$.

2.1 Mean, variance, and generating function

For $X\sim\operatorname{Poisson}(\mu)$,

\[ \boxed{E(X)=\mu,\qquad \operatorname{Var}(X)=\mu.} \]

An efficient derivation uses the factorial moment:

\[ \begin{aligned} E\{X(X-1)\} &=\sum_{x=2}^{\infty}x(x-1)e^{-\mu}\frac{\mu^x}{x!}\\ &=\mu^2\sum_{j=0}^{\infty}e^{-\mu}\frac{\mu^j}{j!} =\mu^2. \end{aligned} \]

Because $X^2=X(X-1)+X$, $E(X^2)=\mu^2+\mu$ and hence $\operatorname{Var}(X)=\mu$. The probability-generating function is

\[ G_X(s)=E(s^X)=\exp\{\mu(s-1)\}. \]

Equality of mean and variance is a model implication, not a universal law for count data.

3 From Rare Trials to Poisson Counts

Let $X_n\sim\operatorname{Binomial}(n,p_n)$, with $n\to\infty$, $p_n\to0$, and $np_n\to\mu$. For fixed $x$,

\[ \begin{aligned} P(X_n=x) &={n\choose x}p_n^x(1-p_n)^{n-x}\\ &=\frac{n(n-1)\cdots(n-x+1)}{n^x x!} (np_n)^x(1-p_n)^n(1-p_n)^{-x}\\ &\longrightarrow e^{-\mu}\frac{\mu^x}{x!}. \end{aligned} \]

The limit explains the Poisson approximation to many opportunities for rare, approximately independent events. It does not justify a Poisson model when opportunities cluster, probabilities differ substantially, or one event changes the chance of another.

4 The Homogeneous Poisson Process

A counting process $\{N(t):t\geq0\}$ is a homogeneous Poisson process with rate $\lambda>0$ if:

$N(0)=0$;
counts in disjoint intervals are independent;
increments are stationary and

\[ N(t+h)-N(t)\sim\operatorname{Poisson}(\lambda h). \]

Consequently,

\[ N(t)\sim\operatorname{Poisson}(\lambda t), \qquad E\{N(t)\}=\operatorname{Var}\{N(t)\}=\lambda t. \]

The rate $\lambda$ has units of events per unit time, distance, area, or other exposure. The product $\lambda t$ is the expected number of events in a region of size $t$.

An equivalent local construction assumes, for small $h$,

\[ P\{N(t+h)-N(t)=1\}=\lambda h+o(h), \]

\[ P\{N(t+h)-N(t)\geq2\}=o(h), \]

together with independent, stationary increments. These conditions say that events occur singly at a stable rate and have no history-dependent clustering.

4.1 Interarrival times

Let $T_1$ be the time to the first event. The events $\{T_1>t\}$ and $\{N(t)=0\}$ are identical, so

\[ P(T_1>t)=P\{N(t)=0\}=e^{-\lambda t}. \]

Thus $T_1\sim\operatorname{Exponential}(\lambda)$. Independent increments imply that successive interarrival times are independent exponential variables. The time $T_k$ of the $k$th event is their sum and has a gamma distribution with shape $k$ and rate $\lambda$.

This count–waiting-time duality links the process directly to exponential and geometric distributions.

4.2 Superposition, thinning, and conditioning on the total

Poisson processes have useful closure properties:

Superposition: independent processes with rates $\lambda_1,\ldots,\lambda_K$ combine into a Poisson process with rate $\sum_k\lambda_k$.
Thinning: independently retaining each event with probability $p$ produces a Poisson process of rate $p\lambda$; retained and discarded processes are independent.
Conditional locations: given $N(t)=n$, the $n$ unordered event locations are independent uniform points on $(0,t)$; their ordered locations are the corresponding order statistics.

For independent counts $X_k\sim\operatorname{Poisson}(\mu_k)$,

\[ \sum_{k=1}^KX_k\sim\operatorname{Poisson}\left(\sum_{k=1}^K\mu_k\right). \]

Conditioning on their total gives

\[ (X_1,\ldots,X_K)\mid \sum_kX_k=n \sim\operatorname{Multinomial}\left(n,\frac{\mu_1}{\mu_+},\ldots,\frac{\mu_K}{\mu_+}\right), \]

where $\mu_+=\sum_k\mu_k$. This connects unconditional event totals to the fixed-total allocation model in categorical and multinomial distributions.

5 Rates, Exposure, and Estimation

Suppose independent counts satisfy

\[ Y_i\sim\operatorname{Poisson}(\lambda t_i), \]

where $t_i$ is known exposure. The likelihood for a common rate is

\[ L(\lambda;y)\propto \lambda^{\sum_i y_i}\exp\left(-\lambda\sum_i t_i\right) \prod_i t_i^{y_i}. \]

Differentiating the log-likelihood gives

\[ \boxed{\hat\lambda=\frac{\sum_i y_i}{\sum_i t_i}.} \]

The numerator is an event count and the denominator is total exposure. Comparing raw counts when exposures differ confounds rate with opportunity. In regression, $\log t_i$ is commonly used as an offset so the model targets an event rate rather than an unadjusted count.

6 Worked Example: Sequence Starts Along a Genome

Consider an idealized library in which read starts are placed independently and uniformly along a genome of length $G$. If there are $m$ starts and a region has length $L\ll G$, its start count is binomial with probability approximately $L/G$. When $m$ is large, $L/G$ is small, and $mL/G\to\lambda L$, the count is approximately

\[ N(L)\sim\operatorname{Poisson}(\lambda L). \]

Under the corresponding homogeneous process, the expected number of starts is $\lambda L$, the variance is also $\lambda L$, and the probability of no start in the region is $e^{-\lambda L}$. Distances between successive starts are exponential with mean $1/\lambda$.

This is a sampling model, not a guarantee about real sequencing. GC bias, mappability, fragmentation preferences, duplicate molecules, and excluded regions make rates nonconstant or events dependent. The read start is the point-process event, while the independently sampled organism or specimen remains the experimental unit for biological inference. More reads characterize one library more precisely; they do not create more biological replicates.

7 When the Poisson Model Fails

Model implication	Failure mechanism	Diagnostic or response
Constant rate	Rate changes with time, position, covariates, or batch	Stratify or fit a nonhomogeneous process/regression model
Independent increments	Bursts, inhibition, duplicate events, or contagion	Inspect inter-event structure; use cluster, renewal, or self-exciting models
Mean equals variance	Unmodeled heterogeneity or dependence	Compare conditional mean and variance; consider negative-binomial or hierarchical models
No extra structural zeros	Inactive units or detection failure	Model the observation process; assess hurdle or zero-inflated alternatives
Exposure is known and comparable	Different region lengths, person-time, or library opportunities	Include exposure explicitly, often as an offset

Overdispersion is common when units have different latent rates. If $Y\mid\Lambda\sim\operatorname{Poisson}(\Lambda)$, then

\[ \operatorname{Var}(Y)=E(\Lambda)+\operatorname{Var}(\Lambda), \]

which exceeds the mean whenever the rate varies. Underdispersion can arise from inhibition or fixed-spacing mechanisms. Neither pattern is repaired by treating more nested technical observations as independent.

8 Common Pitfalls

Using Poisson for every nonnegative count: support alone does not supply constant-rate or independence assumptions.
Confusing $\lambda$ with $\lambda t$: one is a rate with units; the other is an expected count.
Ignoring exposure: equal counts over unequal opportunities imply unequal rates.
Calling mean–variance equality an empirical identity: it is conditional on the Poisson model.
Inferring independence from a Poisson-looking histogram: marginal counts can look Poisson even when event locations are dependent.
Treating sequence reads or cells as biological replicates: point-process events and experimental units answer different scientific questions.

9 Connections to the Knowledge Map

Bernoulli and binomial distributions supplies the rare-event limit.
Exponential and geometric distributions develops waiting times and memorylessness.
Categorical and multinomial distributions describes Poisson counts conditional on their total.
Expectation, variance, and moments supports factorial moments and mixture variance.
Independence and conditional independence explains the process factorization.

10 Take-home Messages

The Poisson distribution models a count with mean and variance $\mu$ under a specific event-generating mechanism.
A homogeneous Poisson process has independent, stationary increments and constant event rate $\lambda$.
Poisson counts, exponential interarrival times, superposition, thinning, and conditional multinomial allocation are parts of one coherent structure.
Rates must be interpreted relative to exposure.
Heterogeneity, clustering, inhibition, excess zeros, and measurement bias can invalidate a simple Poisson model.
Technical events such as reads do not replace independent biological samples.

11 References

Kingman, J. F. C. Poisson Processes. Oxford University Press, 1993. https://doi.org/10.1093/oso/9780198536932.001.0001.
Ross, S. M. Introduction to Probability Models, 12th ed. Academic Press, 2019.
Johnson, N. L., Kemp, A. W., and Kotz, S. Univariate Discrete Distributions, 3rd ed. Wiley, 2005. https://doi.org/10.1002/0471715816.
Lander, E. S. and Waterman, M. S. “Genomic Mapping by Fingerprinting Random Clones: A Mathematical Analysis.” Genomics 2(3), 1988, 231–239. https://doi.org/10.1016/0888-7543(88)90007-9.
R Core Team. “The Poisson Distribution.” R Statistical Functions Documentation. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Poisson.html.

--- title: "Poisson Distribution and Poisson Processes" subtitle: "Counts, event rates, and random arrivals" description: "The Poisson distribution and homogeneous Poisson process, including derivations, waiting-time relationships, assumptions, and biological interpretation." author: "Jiahong Sun" date: "2026-09-11" date-modified: "2026-09-11" categories: [Statistics, Probability, Distributions, Stochastic Processes] toc: true number-sections: true draft: false --- # Topic at a Glance > **One-sentence summary:** The Poisson distribution models the number of events in a fixed exposure, while a Poisson process models how such events accumulate through continuous time or space at a constant rate. | Item | Description | |---|---| | Domain | Probability & Randomness; count distributions and point processes | | Main question | How many events occur in a region, and where or when do they occur? | | Support | Count $X\in\{0,1,2,\ldots\}$; process $N(t)\in\{0,1,2,\ldots\}$ | | Parameters | Mean count $\mu>0$; process rate $\lambda>0$ per unit exposure | | Typical applications | Mutation counts, sequence starts, recombination events, failures, arrivals | | Prerequisites | [Bernoulli and binomial distributions](bernoulli-and-binomial-distributions.qmd); [independence and conditional independence](independence-and-conditional-independence.qmd) | # The Poisson Distribution A random variable $X$ has a Poisson distribution with mean parameter $\mu>0$ when $$ \boxed{ P(X=x)=e^{-\mu}\frac{\mu^x}{x!}, \qquad x=0,1,2,\ldots } $$ written $X\sim\operatorname{Poisson}(\mu)$. The parameter is dimensionless: if events occur at rate $\lambda$ per unit exposure and the exposure is $t$, then $\mu=\lambda t$. Normalization follows from the exponential series: $$ \sum_{x=0}^{\infty}e^{-\mu}\frac{\mu^x}{x!} =e^{-\mu}e^\mu=1. $$ The probability of zero events is $e^{-\mu}$, and successive probabilities obey $$ \frac{P(X=x+1)}{P(X=x)}=\frac{\mu}{x+1}. $$ This recursion shows that probability masses increase while $x+1<\mu$ and decrease after $x+1>\mu$. ## Mean, variance, and generating function For $X\sim\operatorname{Poisson}(\mu)$, $$ \boxed{E(X)=\mu,\qquad \operatorname{Var}(X)=\mu.} $$ An efficient derivation uses the factorial moment: $$ \begin{aligned} E\{X(X-1)\} &=\sum_{x=2}^{\infty}x(x-1)e^{-\mu}\frac{\mu^x}{x!}\\ &=\mu^2\sum_{j=0}^{\infty}e^{-\mu}\frac{\mu^j}{j!} =\mu^2. \end{aligned} $$ Because $X^2=X(X-1)+X$, $E(X^2)=\mu^2+\mu$ and hence $\operatorname{Var}(X)=\mu$. The probability-generating function is $$ G_X(s)=E(s^X)=\exp\{\mu(s-1)\}. $$ Equality of mean and variance is a model implication, not a universal law for count data. # From Rare Trials to Poisson Counts Let $X_n\sim\operatorname{Binomial}(n,p_n)$, with $n\to\infty$, $p_n\to0$, and $np_n\to\mu$. For fixed $x$, $$ \begin{aligned} P(X_n=x) &={n\choose x}p_n^x(1-p_n)^{n-x}\\ &=\frac{n(n-1)\cdots(n-x+1)}{n^x x!} (np_n)^x(1-p_n)^n(1-p_n)^{-x}\\ &\longrightarrow e^{-\mu}\frac{\mu^x}{x!}. \end{aligned} $$ The limit explains the Poisson approximation to many opportunities for rare, approximately independent events. It does not justify a Poisson model when opportunities cluster, probabilities differ substantially, or one event changes the chance of another. # The Homogeneous Poisson Process A counting process $\{N(t):t\geq0\}$ is a homogeneous Poisson process with rate $\lambda>0$ if: 1. $N(0)=0$; 2. counts in disjoint intervals are independent; 3. increments are stationary and $$ N(t+h)-N(t)\sim\operatorname{Poisson}(\lambda h). $$ Consequently, $$ N(t)\sim\operatorname{Poisson}(\lambda t), \qquad E\{N(t)\}=\operatorname{Var}\{N(t)\}=\lambda t. $$ The rate $\lambda$ has units of events per unit time, distance, area, or other exposure. The product $\lambda t$ is the expected number of events in a region of size $t$. An equivalent local construction assumes, for small $h$, $$ P\{N(t+h)-N(t)=1\}=\lambda h+o(h), $$ $$ P\{N(t+h)-N(t)\geq2\}=o(h), $$ together with independent, stationary increments. These conditions say that events occur singly at a stable rate and have no history-dependent clustering. ## Interarrival times Let $T_1$ be the time to the first event. The events $\{T_1>t\}$ and $\{N(t)=0\}$ are identical, so $$ P(T_1>t)=P\{N(t)=0\}=e^{-\lambda t}. $$ Thus $T_1\sim\operatorname{Exponential}(\lambda)$. Independent increments imply that successive interarrival times are independent exponential variables. The time $T_k$ of the $k$th event is their sum and has a gamma distribution with shape $k$ and rate $\lambda$. This count--waiting-time duality links the process directly to [exponential and geometric distributions](exponential-and-geometric-distributions.qmd). ## Superposition, thinning, and conditioning on the total Poisson processes have useful closure properties: - **Superposition:** independent processes with rates $\lambda_1,\ldots,\lambda_K$ combine into a Poisson process with rate $\sum_k\lambda_k$. - **Thinning:** independently retaining each event with probability $p$ produces a Poisson process of rate $p\lambda$; retained and discarded processes are independent. - **Conditional locations:** given $N(t)=n$, the $n$ unordered event locations are independent uniform points on $(0,t)$; their ordered locations are the corresponding order statistics. For independent counts $X_k\sim\operatorname{Poisson}(\mu_k)$, $$ \sum_{k=1}^KX_k\sim\operatorname{Poisson}\left(\sum_{k=1}^K\mu_k\right). $$ Conditioning on their total gives $$ (X_1,\ldots,X_K)\mid \sum_kX_k=n \sim\operatorname{Multinomial}\left(n,\frac{\mu_1}{\mu_+},\ldots,\frac{\mu_K}{\mu_+}\right), $$ where $\mu_+=\sum_k\mu_k$. This connects unconditional event totals to the fixed-total allocation model in [categorical and multinomial distributions](categorical-and-multinomial-distributions.qmd). # Rates, Exposure, and Estimation Suppose independent counts satisfy $$ Y_i\sim\operatorname{Poisson}(\lambda t_i), $$ where $t_i$ is known exposure. The likelihood for a common rate is $$ L(\lambda;y)\propto \lambda^{\sum_i y_i}\exp\left(-\lambda\sum_i t_i\right) \prod_i t_i^{y_i}. $$ Differentiating the log-likelihood gives $$ \boxed{\hat\lambda=\frac{\sum_i y_i}{\sum_i t_i}.} $$ The numerator is an event count and the denominator is total exposure. Comparing raw counts when exposures differ confounds rate with opportunity. In regression, $\log t_i$ is commonly used as an offset so the model targets an event rate rather than an unadjusted count. # Worked Example: Sequence Starts Along a Genome Consider an idealized library in which read starts are placed independently and uniformly along a genome of length $G$. If there are $m$ starts and a region has length $L\ll G$, its start count is binomial with probability approximately $L/G$. When $m$ is large, $L/G$ is small, and $mL/G\to\lambda L$, the count is approximately $$ N(L)\sim\operatorname{Poisson}(\lambda L). $$ Under the corresponding homogeneous process, the expected number of starts is $\lambda L$, the variance is also $\lambda L$, and the probability of no start in the region is $e^{-\lambda L}$. Distances between successive starts are exponential with mean $1/\lambda$. This is a sampling model, not a guarantee about real sequencing. GC bias, mappability, fragmentation preferences, duplicate molecules, and excluded regions make rates nonconstant or events dependent. The read start is the point-process event, while the independently sampled organism or specimen remains the experimental unit for biological inference. More reads characterize one library more precisely; they do not create more biological replicates. # When the Poisson Model Fails | Model implication | Failure mechanism | Diagnostic or response | |---|---|---| | Constant rate | Rate changes with time, position, covariates, or batch | Stratify or fit a nonhomogeneous process/regression model | | Independent increments | Bursts, inhibition, duplicate events, or contagion | Inspect inter-event structure; use cluster, renewal, or self-exciting models | | Mean equals variance | Unmodeled heterogeneity or dependence | Compare conditional mean and variance; consider negative-binomial or hierarchical models | | No extra structural zeros | Inactive units or detection failure | Model the observation process; assess hurdle or zero-inflated alternatives | | Exposure is known and comparable | Different region lengths, person-time, or library opportunities | Include exposure explicitly, often as an offset | Overdispersion is common when units have different latent rates. If $Y\mid\Lambda\sim\operatorname{Poisson}(\Lambda)$, then $$ \operatorname{Var}(Y)=E(\Lambda)+\operatorname{Var}(\Lambda), $$ which exceeds the mean whenever the rate varies. Underdispersion can arise from inhibition or fixed-spacing mechanisms. Neither pattern is repaired by treating more nested technical observations as independent. # Common Pitfalls 1. **Using Poisson for every nonnegative count:** support alone does not supply constant-rate or independence assumptions. 2. **Confusing $\lambda$ with $\lambda t$:** one is a rate with units; the other is an expected count. 3. **Ignoring exposure:** equal counts over unequal opportunities imply unequal rates. 4. **Calling mean--variance equality an empirical identity:** it is conditional on the Poisson model. 5. **Inferring independence from a Poisson-looking histogram:** marginal counts can look Poisson even when event locations are dependent. 6. **Treating sequence reads or cells as biological replicates:** point-process events and experimental units answer different scientific questions. # Connections to the Knowledge Map - [Bernoulli and binomial distributions](bernoulli-and-binomial-distributions.qmd) supplies the rare-event limit. - [Exponential and geometric distributions](exponential-and-geometric-distributions.qmd) develops waiting times and memorylessness. - [Categorical and multinomial distributions](categorical-and-multinomial-distributions.qmd) describes Poisson counts conditional on their total. - [Expectation, variance, and moments](expectation-variance-and-moments.qmd) supports factorial moments and mixture variance. - [Independence and conditional independence](independence-and-conditional-independence.qmd) explains the process factorization. # Take-home Messages 1. The Poisson distribution models a count with mean and variance $\mu$ under a specific event-generating mechanism. 2. A homogeneous Poisson process has independent, stationary increments and constant event rate $\lambda$. 3. Poisson counts, exponential interarrival times, superposition, thinning, and conditional multinomial allocation are parts of one coherent structure. 4. Rates must be interpreted relative to exposure. 5. Heterogeneity, clustering, inhibition, excess zeros, and measurement bias can invalidate a simple Poisson model. 6. Technical events such as reads do not replace independent biological samples. # References - Kingman, J. F. C. *Poisson Processes*. Oxford University Press, 1993. [https://doi.org/10.1093/oso/9780198536932.001.0001](https://doi.org/10.1093/oso/9780198536932.001.0001). - Ross, S. M. *Introduction to Probability Models*, 12th ed. Academic Press, 2019. - Johnson, N. L., Kemp, A. W., and Kotz, S. *Univariate Discrete Distributions*, 3rd ed. Wiley, 2005. [https://doi.org/10.1002/0471715816](https://doi.org/10.1002/0471715816). - Lander, E. S. and Waterman, M. S. “Genomic Mapping by Fingerprinting Random Clones: A Mathematical Analysis.” *Genomics* 2(3), 1988, 231–239. [https://doi.org/10.1016/0888-7543(88)90007-9](https://doi.org/10.1016/0888-7543(88)90007-9). - R Core Team. “The Poisson Distribution.” *R Statistical Functions Documentation*. [https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Poisson.html](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Poisson.html).